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We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $aleph_{alpha}$, then the set has size $aleph_{alpha}$ for any regular $aleph_{alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworths decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.
In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a
By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]={1,2,ldots, n}$ for some integers $nge kge 1$, and in which two such sets are adjacent if and only if they realise a certain order type specified in ad
A signed graph is a pair $(G, sigma)$, where $G$ is a graph and $sigma: E(G) to {+, -}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular colorin
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $phicolon V(G)to mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $phi(u)$ is different from the colour $sigma(uv)phi(v
Fix a countable nonstandard model $mathcal M$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $mathcal N succ mathcal M$ that are allowed, we still find that there are $2^{aleph_0}$ pos